Although it is quite remarkable that such equations can be solved analytically, it is important to note that most of these solutions have been found in the two flavors case, where three degrees of freedom are present. Building solutions with \(N>2\)  is a more complicated task, as can be seen in Refs. [1]}, [2]} and [3]}. The main goal of this work is to construct analytical solutions of the Einstein-NLSM theory for arbitrary values of \(N\) . In particular, we will construct solutions that describe black holes and black strings by combining the generalized hedgehog ansatz with the Euler angle parameterization through the maximal embbeding of \(SU(2)\) . This maximal embedding, introduced in Refs. [4]}, [5]} and [6]}, allows us to construct in a direct way, genuine \(SU(N)\)  fields that are not trivial embeddings of \(SU(2)\)  into \(SU(N)\) . The power of this formalism can be seen in Refs. [7]}, [8]}, [9]} and [10]}, where nuclear pasta states in the \(SU(N)\) -Skyrme model
and black holes in the Einstein \(SU(N)\) -Yang-Mills theory have been constructed.
We will see that the pionic black objects constructed here possess a non-trivial thermodynamics and intriguing geometrical properties. In this way, we will generalize to the \(SU(N)\)  case the solutions constructed in Refs. [11]}, [12]} and [13]} for the NLSM, explicitly showing the role that the flavor number plays both in the thermodynamics and in the geometry of these configurations.
